Copying these fluffy clouds?
Well, I eventually figured it out.

Not quite sure what I was doing wrong, but I'll say what I did and post the .blend file.

Here's the image I ended up with. 



It's not perfect, mainly the lighting I think, but I'm pretty happy. It matches the 'cartoony' cloud style I was going for.

I started out just making a simple cloud-shape out of metaballs.

I then more or less copied the node setup above, what I actually used being below:



I then added a subsurf modifier, set to 3, and two displacement modifiers, shown below.



For both displacements, I used an empty called iCloud to control the coordinates. The first texture was a Voronoi, shown below.



The second was a Clouds texture, /.



I then had two suns, one with a strength of 8, and a very slight warm colour. The second was mildly blue, with a strength of 2. They are both pointing down, slighty angled from different sides.

This was my final scene before rendering.



And that's how I ended up with my render! Might be a bit of a drawn out answer, but I know how long I spent desperately trying to find answers, so I wanted to help out anyone else as much as I could.

EDIT: Here is the .blend file.